Observation-Constrained Hidden-Set Decisions: Exact Bayes Values, Adaptivity, and Coverage

This Version 2 working paper studies a finite Bayesian decision problem in which a learner must name one member of a hidden subset. It separates two information constraints: a free finite-alphabet encoder, which represents communication capacity, and a physical protocol that can only query whether named points belong to the hidden set. For any rational prior, the optimal deterministic value of K free messages is weighted maximum K-point coverage. More importantly, after I perfect point-membership inspections, both batch and deterministic adaptive protocols have exact value equal to weighted maximum (I+1)-point coverage; adaptivity gives no advantage for this observation family and one-hit loss. With reliable whole-output delivery, the value interpolates exactly between the prior one-guess value and this physical ceiling. In the uniform (R,M,I,rho)=(8,2,2,3/4) instance, the physical value is 61/112, strictly below the unrestricted binary-transcript capacity value 73/112. The paper also gives the uniform binomial specialization, rational stochastic robustness, a simple 1-design characterization of decision-neutral records, deterministic data processing, a g-vulnerability bridge, and exact finite values for several multiple-answer reward families. Lean 4 machine-checks the stated finite results with exact rational probabilities and explicit witnesses. The contribution is scoped to the declared finite communication and perfect point-inspection models. It does not claim a new maximum-coverage algorithm, a general no-adaptivity theorem, noisy or pooled testing results, a strategic signaling equilibrium, a full Werewolf-game value, or an agent-performance score.